A Simple Proof for the Theorems of Pascal and Pappus

Marian Palej

Abstract: The theorem of Pascal concerning a hexagon inscribed in a conic is very useful in many geometrical constructions and ought to be included in a normal course on descriptive geometry. Though the citation of this theorem is possible in a short lecture, its proof is very often omitted due to a lack of time and because students at technical universities have no basic knowledge in projective geometry. As far as we know, this discipline is not contained in the curricula of technical universities. However, a lecture without proofs is incomplete and satisfies neither lecturers nor students. Therefore the author presents a proof of Pascal's theorem which does not require any knowledge of projective geometry. The conic is seen as the contour of a quadric $\Phi$, and some pairs of lines define conical surfaces $\Gamma_1, \Gamma_2$. Then the intersections between these three quadrics $\Phi, \Gamma_1, \Gamma_2$ lead to three collinear Pascal's points. When the quadric $\Phi$ is replaced by a conical surface $\Gamma_3$ the analysis of intersections between the three surfaces leads to an immediate proof of Pappus theorem.